1) Find the Rank Dependent Utility and the Certainty Equivalent of the lottery (80*,*) under *g*(*p*) = *p*^{2 }and *u*(*x*) = *x*.

2) __RDU as a better descriptive model: __Start with the famous Allais Paradox of

01) and (5 89). Let

*u*(0) = 0. Show that under EU, one would get *.*11*u*(1*m*) *> .*10*u*(5*m*) and *.*11(1*m*) *< .*10*u*(5*m*) respectively, which cannot hold simultaneously. Next, show that under RDU with *g*(*p*) = *p*^{2}, show that these two conditions become and . Show that these two conditions could

hold simultaneuosly as * ^{.}*.

__3) Independence Axiom and Reduction of Compound Lotteries:__Suppose the decision-maker is indifferent between two lotteries *L*_{1 }= {10*,.*5;0*,.*5} and *L*_{2 }= {4*,*1}. Show that under the independence axiom she should also be indifferent between *L*_{1 }and {10*,.*25;4*,.*5;0*,.*25}. (Hint: Start with *L*_{1 }∼ *L*_{2}*, *then using the independence axiom mix *.*5 proportion of both sides with *.*5 proportion of *L*_{1}. What do you get?)

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4) Assume that *S *= {*s*_{1}*,s*_{2}*,s*_{3}}, *u*(*x*) = *x*, *W*(*s*_{1}) = *W*(*s*_{2}) = *W*(*s*_{3}) = 0*.*4, and *W*(*s*_{1}*,s*_{2}) = *W*(*s*_{1}*,s*_{3}) = *W*(*s*_{2}*,s*_{3}) = 0*.*6. Calculate the RDU value of the prospect (1*,s*_{1};0*,s*_{2};9*,s*_{3}).